Pablo Lummerzheim | Searcharxiv

arXiv2026

Hecke Algebras for Minimal Equicontinuous Cantor Actions

In this work we extend the concept of Hecke algebras to a pair of a separable infinite profinite group and a closed core-free subgroup. Motivated by applications to minimal equicontinuous Cantor actions, which are characterized by such pairs, we characterize isometric *-algebra isomorphisms between such Hecke algebras by showing that they always arise as Koopman operators of homeomorphisms between the double coset spaces combined with the multiplication by some character. We then prove that two minimal equicontinuous Cantor actions with isometric isomorphic Hecke algebras have the same Steinitz order. We provide examples including the infinite dihedral group and the Grigorchuk group.

math.DS↗

arXiv2026

Strict irreducibility of Markov chains and ergodicity of skew products

We consider a family of measure preserving transformations, which act on a common probability space and are chosen at random by a stationary ergodic Markov chain. This setting defines an instance of a random dynamical system (RDS), which may be described in terms of a step skew product. In many contexts it is desirable to know whether ergodicity of the family implies ergodicity of the skew product. Introducing the notion of strict irreducibility for Markov kernels we shall characterize the class of Markov chains for which the aforementioned implication holds true. We thereby extend a sufficient condition of Bufetov for the case of finite state Markov chains to general state spaces and show that it is in fact also necessary. As an application we obtain an explicit description of the limit in ergodic theorems for a suitable class of random transformations.

math.DS↗